# A Segmentation Approach for Stochastic Geological Modeling Using Hidden Markov Random Fields

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## Abstract

Stochastic modeling methods and uncertainty quantification are important tools for gaining insight into the geological variability of subsurface structures. Previous attempts at geologic inversion and interpretation can be broadly categorized into geostatistics and process-based modeling. The choice of a suitable modeling technique directly depends on the modeling applications and the available input data. Modern geophysical techniques provide us with regional data sets in two- or three-dimensional spaces with high resolution either directly from sensors or indirectly from geophysical inversion. Existing methods suffer certain drawbacks in producing accurate and precise (with quantified uncertainty) geological models using these data sets. In this work, a stochastic modeling framework is proposed to extract the subsurface heterogeneity from multiple and complementary types of data. Subsurface heterogeneity is considered as the “hidden link” between multiple spatial data sets. Hidden Markov random field models are employed to perform three-dimensional segmentation, which is the representation of the “hidden link”. Finite Gaussian mixture models are adopted to characterize the statistical parameters of multiple data sets. The uncertainties are simulated via a Gibbs sampling process within a Bayesian inference framework. The proposed modeling method is validated and is demonstrated using numerical examples. It is shown that the proposed stochastic modeling framework is a promising tool for three-dimensional segmentation in the field of geological modeling and geophysics.

### Keywords

Geological modeling Geostatistics Uncertainty quantification Gibbs sampling Heterogeneity## Notes

### Acknowledgements

Hui Wang and Florian Wellmann would like to acknowledge the support from the German research foundation (DFG) through the Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH Aachen University. The authors would like to thank the anonymous reviewers for their constructive comments that have helped to improve the paper significantly.

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